Applications of Topology Optimization Techniques in Seismic Design of Structure - Structural Seismic Design Optimization and Earthquake Engineering - page 248

∑ ω 2

Ω

f = − (1.59)

Example 3: Planar Frame
with Non-Structural Mass

j

j J ∈

in which Ω is the excitation frequency and
J

A three-level planar frame is considered with 6
non-structural masses of 20,000kg each attached
to it as shown in Figure 8a. The ratio between the
stiffness of solid and void areas is reduced to
E : = 100 . All other parameters are similar to
example 1.
We fist maximize the fundamental frequency
of the frame. The evolution history of the funda-
mental frequency and the final solution are shown
in Figures 8b and 8c respectively. The fundamen-
tal frequency has increased by 85% from 40.2 to
74.6rad/s.

⊆ {

}

1, , is a set of natural frequencies
considered. If we only consider the closest natu-
ral frequencies to Ω, the problem reduces to
maximizing the gap between the adjacent natural
frequencies. Maximizing the fundamental fre-
quency is a special case of this problem with Ω
= 0.
Maximizing the gap between two natural fre-
quencies can lead to multiple eigenfrequencies
(Du and Olhoff 2007). The following example
illustrates this.

N d

Table 1. Comparison of the results of the three approaches used to solve example 2

Approach ω 1 (rad/s) ω 2 (rad/s) ω 3 (rad/s)

Using single eigenvalue sensitivities (1.18) 257.1 267.1 330.9

Using the mean eigenvalue (1.33) as objective function 248.0 424.4 540.2

Using multiple eigenvalue sensitivities (1.43) 273.5 284.0 337.6

Figure 8. Example 3: initial design (a), final solution (b), and evolution of the fundamental frequency (c)